Representation of Jacobi polynomials

["Daniel Lozier" <lozier@nist.gov>]

Sender: thk@science.uva.nl (Tom Koornwinder)
Subject: sci.math.research question


>From: "T.H. Koornwinder" <sfmkoo01@newton.cam.ac.uk>
Newsgroups: sci.math.research
Subject: Re: Representation of Jacobi polynomials
Date: 8 May 2001 00:20:01 -0500
Organization: University of Cambridge, England

Roland Franzius <Roland.Franzius@uos.de> wrote to sci.math.research:

> In Gustafson, Partial Differential Equations, I just found a formula
>
> JacobiP(n,1/2,1/2,x) ~ 1/Sqrt(1-x^2) Sin((n+1) ArcCos(x))
>
> Is there a generalization of this formula for all alpha, beta?

and Pierre Asselin (lpa@rexx.com) replied:

> The Jacobis(1/2,1/2) are Chebychev polynomials of the second kind,
> hence the formula.  I don't think it generalizes.
>
> Well, the Jacobi polynomials are hypergeometric functions.  Maybe you
> can derive something useful from that fact.  See Abramowitz and
Stegun.


A possible generalization to the case of Gegenbauer (ultraspherical)
polynomials, i.e. Jacobi polynomials P_n^{(a,b)}(x) with a=b,
has the form

(1)  (sin x)^{(2a)} P_n^{(a,a)}(cos x)=sum_{k=0}^infinity c_k
sin((n+2k+1)x).

If a=1/2 then c_k vanishes for k>0.
The explicit values of c_k can be found by combining the
following formulas in Abramowitz and Stegun:
8.7.1 (trigonometric expansion of Legendre function),
22.5.60 (ultraspherical polynomial expressed as Legendre function),
22.5.20 (Jacobi polynomial expressed as ultraspherical polynomial).

If we write the right-hand side of (1) as

sum_{k=0}^infinity c_k/(2i) e^{i(n+2k+1)x}-
sum_{k=0}^infinity c_k/(2i) e^{-i(n+2k+1)x}

(essentially 8.3.2, 8.3.3 in A&S)
then each of the two sums is also a solution of the second order
differential equation

f''(x) + (1-2a) cot(x)f'(x) + (n+2a) f(x)=0

for f(x)=(sin x)^{(2a)} P_n^{(a,a)}(cos x).
In the case a=1/2 we thus get a linear combinatation of e^{i(n+1)x}
and e^{-i(n+1)x}.


q-ultraspherical polynomials C_n(x; beta | q) (see the book
Gasper & Rahman, Basic hypergeometric series, Section 7.4), still
have the property that

C_n(cos x; q | q) = const. sin((n+1)x) / sin x.

Formula (1) also generalizes explicitly in that case, if we replace
the function cos x |-> (sin x)^{(2a)} by the weight function in
the orthogonality relations for the q-ultraspherical plynomials.
The resulting sine expansion is formula (7.4.8) in Gasper & Rahman.


                    Tom Koornwinder
                    KdV Institute, Univ. of Amsterdam
                    and visiting Isaac Newton Institute, Cambridge, UK

                    email: thk@science.uva.nl